May and Haldane is an important contribution summarising a programme of research that the Bank of England had been developing in collaboration with Lord May. However, just under a third of the paper, the section entitled Potential causes of an initial shock focuseson a paper, Eroding market stability by proliferation of financial instruments written by three physicists, Caccioli, Marsili and Vivo, and published in the European Physics Journal B in 2009. May and Haldane describe the paper as

A sophisticated and important analysis of a major flaw in the pricing of derivatives.

The purpose of this post is to describe why the analysis undertaken by the physicists Caccioli, Marsili and Vivo is deeply flawed and goes on to discuss how damaging "science based policy advice" can, consequently, be deeply flawed if it lacks a scientific base but is propagated by scientific authority. This is presented as a concrete counter-example to the opinions offered Brian Cox and Robin Ince on the need to place science at the pinnacle of policy advice.

Caccioli et al (I discuss this published paper, a pre-print is publicly available on arXiv) argue that

Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently provides more means for risk diversification, making the market more efficient and complete. In the simple market of interacting traders discussed here, the proliferation of financial instruments erodes systemic stability and it drives the market to a critical state characterized by large susceptibility, strong fluctuations and enhanced correlations among risks. This suggests that the hypothesis of APT may not be compatible with a stable market dynamics. In this perspective, market stability acquires the properties of a common good, which suggests that appropriate measures should be introduced in derivative markets, to preserve stability.

By APT the authors mean what mathematicians understand as the Fundamental Theorem of Asset Pricing (FTAP), which I have discussed at length. The issue that Caccioli et al focuses on, and it is an important point that is generally overlooked in economic and financial discussions of the FTAP, is that of market incompleteness.

Central to the FTAP is the following idea: How should you value a bet on roulette before the wheel is spun? In a more general setting, think about a world that jumps from "now" to a future that could take on, randomly, one of K states (in the roulette example K is 36+1 in Europe and 36+2 in the US). The problem finance has is how to price an asset, now, that has a different value in each of the K states, given that we do not know what state will come up. At the core of the FTAP is a simple mathematical result, we can solve a system of simultaneous equations with K unkowns if we have K equations involving the unkowns. This is Cramer's Rule, a mathematical result taught to school kids that has nothing to do with economic or financial theory: if you reject it you may as well reject 2+2=4.

Say the number of assets in a market is N, and each of the N assets has a specific payout in each of the K states of the future world, then if N=K we say that the market is complete. If N>K it means that there are (N-K) "redundant" assets. What this means, as a result of Cramer's Rule, is that these (N-K) redundant assets can be perfectly replicated by constructing a portfolio of the N=K 'primal' assets. In this case, on the basis that we know the initial prices of these N assets we know precisely the prices of the (N-K) remaining assets. Along with this precision comes a cost: we cannot make any profits in the market, the fact that the market is riskless means it cannot generate any profits for the market. This is the basis of the "academic" practice of derivative pricing.

In practice, markets are "incomplete", meaning that N is less than K, assets cannot be priced precisely, there is uncertainty and are risks in the market, which consequently offers (some) profit opportunities. Some economists believe that market incompleteness is about information incompleteness, and so, just as a physicist might repeat an experiment more and more times to get a more and more precise estimate of a parameter, they believe that increasing the number of assets traded in a market brings that market closer to completeness. They treat incompleteness as an epistemological problem, based on limited knowledge, rather than an ontological problem, that the market is either complete or not : one is either a virgin or not, according to conventional science.

Caccioli et al are, rightly, concerned with the fallacy that increasing assets in a market will lead to market completeness. However, rather than writing a short and trivial paper describing why the fallacy is stupid, they seem to buy into the notion that increasing assets in a market could lead to completeness. And in fact they study a finite state market with K=64 which will become complete as soon as the number of assets hits N=64.

I am uploading to SSRN a technical paper describing the numerous conceptual and technical problems I have with Caccioli et al, but here is a non-technical summary.

Caccioli et al do not price assets using the FTAP, rather they assume that financial institutions seek to maximise profit by offering assets where demand is high. The model is naive and despite the claim that the model is dynamic and interacting, it does not attempt to identify a price that matches supply and demand.

Their key result, reported in Haldane and May, is that when N/K>4, the market becomes unstable and volatility explodes. This is Haldane and May's Potential cause of an initial shock.

I do not expect a lay audience to appreciate the fact immediately that if N/K>4 the market is complete, and around 3K of the assets are redundant. All prices are known precisely in the market, so there is no volatility, and the returns to financial institutions offering instruments are zero.

To offer an analogy, the physicists Caccioli et al are arguing that the solar system will become unstable if the planets are moving at 4 times the speed of light. Well a teenager could fantasise about such a result, but policy advisors should not worry too much about it.

The problem seems to be that Caccioli et al do not seem to understand APT/FTAP and take an economic fallacy to show that under a simplistic model of bankers being predatory profit maximisers, the financial system is unstable. The authors bear good scientific reputations, with one, Marsili, being on the editorial board of the European Physics Journal B yet one wonders how such a paper saw the light of day?

Science is littered with examples of rubbish papers going through the peer review process, the MMR controversy originated in a Lacent paper, but the "theory" around science is these errors are quickly corrected by "science". However in the case of Caccioli et al, the error was propagated by even more prestigious scientists, and disseminated, through the BBC, into wider society.

I have a great deal of respect for Lord May (though I might disagree with him, as Denis Mollison has, on the role of uncertainty in biological models) and he should be commended for engaging in finance. He cannot be expected to understand the details of the FTAP. The same could not be said for Dr Haldane.

Haldane (and May?) make the following statement

Caccioli and colleagues note that APT makes several conventional assumptions upon which everything else depends: "perfect competition, market liquidity, no-arbitrage and market completeness". Crucially, this adds up to the implicit assumption that trading activity has no feedback on the dynamical behaviour of markets. And indeed, in the APT-fuelled boomtime that preceded the bust, APT seemed to be very successful. In its imaginary world, market failures are caused by regulatory carelessness, resulting in a focus on creating institutional arrangements that seek to guarantee the premises upon which APT is based. To the contrary, Caccioli and colleagues argued that APT is not a ‘theory’ in the sense habitually used in the sciences, but rather a set of idealized assumptions on which financial engineering is based; that is, APT is part of the problem itself.

Let's dissect this statement in detail.

APT makes several conventional assumptions upon which everything else depends: "perfect competition, market liquidity, no-arbitrage and market completeness". No-arbitrage is essential in the FTAP, but completeness is conditional and will depend on perfect competition (lack of frictions) and market liquidity. These are "conventional" in the sense that they are presented to finance and economics undergraduates, but I teach second year maths students about the problem of incompleteness. Caccioli and colleagues do not have a good enough grasp of the FTAP to comment on its structure.

the implicit assumption that trading activity has no feedback on the dynamical behaviour of markets Neither does the model employed by Caccioli et al, their asset prices do not adjust to balance supply and demand, the most basic example of feedback in markets. Pot calling kettle black?

the APT-fuelled boomtime that preceded the bust Is the BoE claiming that the "boomtime" was a consequence of a mathematical model, was lax interest rate policy and market oversight not more significant? Did the collapse of Bretton woods and fixed exchange rates, stable interest rates and cartel determined commodity prices not have any impact on the development of the derivatives markets that the FTAP was developed in response to?

market failures are caused by regulatory carelessness, resulting in a focus on creating institutional arrangements that seek to guarantee the premises upon which APT is based OK so th eregulation was lax, but it was all the fault of those pesky mathematicians leading us off the straight and narrow. Begs the question, how do you earn your money? Are we forgetting about the fact that the regulators could have argued that Credit Default Swaps, (re-)introduced into the markets in the mid-1990s could have been treated as insurance contracts, and so would not have been tradeable assets, but he regulators chose to let them pass. Are we forgetting the pleas by mathematicians like Phillipe Artzner and Freddy Delbaen or Michael Gordy that fell on deaf ears by regulators and policy makers, see my comment in an earlier post

The problem with modern finance is not in the mathematical models, but in that the models were an end in themselves and not a means for developing a consensus, understanding, knowledge about finance. Banks employed geniuses to develop these models in house that they kept secret, or, they bought black boxes that had been created by geniuses elsewhere. When mathematicians, such as Phillipe Artzner and Freddy Delbaen or Michael Gordy, shone a light on the some of the leading industry models, their illumination was blocked by the towering geniuses, the "masters of the universe", working in banking.

One has the impression that the regulator is looking for a justification for the failure in regulation, and Caccioli et al provides an escape clause: it was the failure of FTAP. The irony is that my interpretation of the FTAP is that it is rooted in reciprocity and emerges out of a philosophical tradition that aimed at establishing Justice rather than to maximise profit, the framework underpinning Caccioli et al, and which I believe is at the heart of the stability problem. But maths is an easier target than mainstream philosophy.

This sorry tale is really about a failure of the model of science that the likes of Brian Cox and Robin Ince adhere to. The Cox-Ince model should prevent the opinions of regulators, that "it was not my fault", or physicists that "bankers are evil" impacting policy advice. But the duo of Haldane and May stand as clear exemplars of the brute fact that we are all human and opinion does trump an abstract notion of the purity of science.

This issue is at the heart of the model of Caccioli et al is that it ignores the basic lack of human objectivity and layers abstract physical analogy on physical analogy rather than consider the market as a social structure. Furthermore, the mess the regulators got themselves into is essentially adhering to the model of science promoted by Cox&Ince, that it is indubitable. Social scientists, on the whole, are far more circumspect, science is socially constructed, and so the problems of subjectivity, and the shakiness of science, need to be taken seriously.

Monday, 7 January 2013

There is a well developed theory that a key impetus for the development of European science in the seventeenth century was magical thinking, developed and promoted through the sixteenth century by the likes of Paracelsus, John Dee and Emperor Rudolph II. While there is little doubt that Hermeticism and Alchemy had a significant influence on the development of natural philosophy, magical thinking cannot explain the uniqueness of the scientific developments in Europe in the 1600s, since magic is a feature of all cultures, notably China. But this factual observation is tempered with an opinion, that good science is open where as magic is hidden and secretive, this is a central theme in Mauss' A General Theory of Magic.

An alternative, minority, theory for the foundations of modern science is in European financial practice. I prefer this theory because, by their very nature markets are social, collaborative, open, forums (those queasy about markets might wish to consider my view). Evidence for the significance of financial practice in the development of science comes in the fact that Copernicus was trained in financial mathematics and wrote on money before he wrote on cosmology, the Merchant Adventurer Thomas Gresham was a more influential contemporary of Dee, despite relative number of contemporary biographies of the two Elizabethians, who laid the basis of the Royal Society with the establishment of Gresham College. Simon Stevin was trained in finance and founded the influential Dutch Mathematical School that inspired Descartes and performed many of the experiments that Galileo is famous for. While Newton's interest in magic has been promoted, the fact that he spent half his life running the Mint is often overlooked.

Furthermore, magic was most influential in central Europe, centred on Rudolph's court, while the scientific revolution was centred in western Europe, by the likes of Huygens and Bernoulli who were as likely to work on financial problems as physical ones. Finally, my observation of good financial practitioners is that, contrary to popular belief, they do not believe they can control the markets, rather they have to navigate through its intrinsic uncertainties using the best tools available - specifically mathematics. This contrasts with the image of the magician controlling nature.

To develop this point, there has been significant criticism of the use of mathematics in the the lead upto the Credit Crisis, that is still affecting all our lives. In particular, the use of a technique known as Value at Risk, to measure the riskiness of investments and the Gaussian Copula, the "formula that killed Wall Street". Now, the fact is that both these techniques emerged out of the investment bank, J.P. Morgan, and the problem in blaming these mathematical techniques for the Credit Crisis is that J.P. Morgan did not engage in trading Credit Default Swaps on Mortgage Backed Securities, and did not hold the alchemical Collatorallised Debt Obligations of Mortgage Backed Securities. The reason? J.P. Morgan having developed the techniques understood them and could not reconcile the model results with their "opinions" of the market. The "science" was being tempered by "opinion" (all this is covered in Gillian Tett's book Fool's Gold, or STS/HPS types might like this).

While, at first sight, these observations might not seem relevant to my criticism of the article by Cox&Ince, I see the two as being linked. Fundamentally, my concern, and it might be unjustified, is that in advocating the primacy of "science" people are endorsing a kind of group think that leads to the sorts of disasters like the Credit Crisis. This might be seen as a bit melodramatic but lets consider two examples, climate change and evolution.

My understanding is that there is little academic dispute about most of the raw data relating to the current climate (modulo the hockey-stick): temperatures are rising and there has been an increase in CO2 in the atmosphere. The academic debate is to what extent man-made emissions of CO2 are causing the rise in temperature, and this is a question of model choice, and since models are made by humans they are, unfortunately for Cox&Ince, social constructs. In November I listened with interest to Andy Kerr talk about plans to bring academics on both sides of the climate debate to discuss the model issues in order to move to a better consensus. This is good science, but the science is not changing as a consequence of new data but as a consequence of human deliberation.

The other great bug-bear of science is evolution. As a mathematician I admire Alan Turing, Turing's most significant work (in terms of citations and impact), undertaken whilst at Manchester at the time of his suicide, was on morphogenisis and this was inspired by Sir D'Arcy Wentworth Thompson'sOn Growth and Form, written to counterbalance the dominance of Darwinian evolution and the emphasis on survival of the fittest. Biological structuralists have come under attack from neo-Darwinists for introducing metaphysical (i.e. mathematical) explanations for phenomena. Who is being un-scientific here, the mathematicians or the neo-Darwinists?

Paul Nurse, the President of the Royal Society whom Jack Stilgoe has associated with Cox&Ince, has argued that "keep science as far as is possible from political, ideological and religious influence". I think Nurse is mistaken here for two reasons.

Firstly, in the thirteenth century the Dominicans, such as Thomas Aquinas and Albert the Great, began Europe's development of science by developing the idea that God could not interfere with nature at will: the deity designed the cosmos, like a machine, but once set in motion they could not interfere with it. The Franciscans, such as John Duns Scotus and William of Ockham, took a different view: God was not constrained by nature. Of course today only a fool would agree with Scotus and Ockham. Or would they? The idea of a Black Swan was developed at this time, the empirical rationalist Dominicans argued that, since no Black Swan had been observed they were not possible. The fideist Fransciscans disagreed, God (and nature) could be capricious and produce a Black Swan if they wanted to, and of course they did. (The Black Swan of Nassim Taleb is different, it is a rare event and in finance we would call Taleb's swan an epistemological problem, Scotus' an ontological problem, and yes finance practitioners are talking in these terms).

The point is, understanding the theological arguments about Black Swans help in our understanding of the contemporary debate. A good scientist, in my view, needs to be humble and accept that they do not, and cannot, know everything. This was central to good financial practice, until "science" became influential through economics, physics and biology.

This leads me on to the second reason why I think Paul Nurse is wrong, by separating themselves from outside influences scientists as susceptible to loosing their humility, or at least of becoming trapped in an ivory tower. This is precisely what happened in finance at those inbstutions that failed, unlike J.P. Morgan. I am of the opinion that the consequences for society could be as dire if science is not integrated with political, ideological and religious influences. Argument should be suppressed if you think you'll lose the argument, but is benign otherwise, something I have discovered as a parent.

Cox&Ince begin their article by describing the significant improvements in the human condition over the past two to three hundred years, which they ascribe to the "scientific method". This brash statement cannot, scientifically, be offered as fact. As a counter example I offer the case of the Soviet Union/Russia. Few would argue that Russia's scientific achievements between 1950 and 2010 were not comparable to those of the UK, France or Germany, but these achievements have not resulted in the quality of life and freedoms enjoyed by western Europeans. Indeed there are many western nations that cannot match Russia's scientific achievement yet are none the less more attractive places to live, Belgium for example.

The argument that science leads to wealth is an opinion not supported by fact, an argument originating in Thomas Gresham's less successful, but more famous, nephew, Francis Bacon. Good science is a consequence of good political structures, and good political structures are closely associated with sound financial practice, and without these social structures there would not have been the technological advances that we enjoy today. This is why I choose Athena as my avatar, acknowledging the precedence of the civilising god over the technological god, Hephaestus. Sorry, am I mixing religion and science here?

Promoting the view that scientists hold a particularly important position in the context of political decision making is dangerous, because despite their best intentions scientists are frequently wrong and are rarely able to distinguish what is true from what is false (the issue about the failure of the Law of Excluded Middle in my last post). A technocracy is just as much an oligarchy as a monarchy, theocracy, plutocracy or stratocracy and is as likely to kill the goose that lays the golden egg as any other system of government that denies dissent.

A historical example of scientists being wrong. Galileo was convicted by the Catholic Inquisition for publishing Dialogue Concerning the Two Chief World Systems in 1632. Galileo's original title for the Dialogue was Dialogue on the Ebb and Flow of the Sea, because, as a consequence of the Copernican theory and mathematics, Galileo argued that there would be one tide a day. When he sent the book to the Church for approval, he was told to change the title because every European sailor knew that there were two tides a day. For the Church of the time, built on Aristotle, Galileo's use of mathematics to describe reality was not just philosophically wrong, it also resulted in absurd conclusions. The historian, Harold Brown explains

Galileo's attempt to account for the tides as a result of the combined daily and annual motion of the earth, and his belief that this argument provided a physical proof that the earth moves, stands as something of an embarrassment.

Are we to consign Galileo to the set of non-scientists and commend the Church for his prosecution, I doubt this would go down well with the likes of Cox&Ince.

The point is scientists must "commit to an evaluative framework", this is what Galileo has done, nailing his colours to mathematics and the Copernican model. The phrase "commit to an evaluative framework" comes out of one of the rare scientific studies of the behaviour of traders, by Daniel Beunza and David Stark, who note that

the trader is emotionally distant from any particular trade, to be able to take a position, the trader must be strongly attached to an evaluative principle and its affiliated instruments

This description of a trader seems to fit that of a scientist. However, what is understood in finance, for example the traders at J.P. Morgan using VaR and the Gaussian Copula, and was understood by Duns Scotus and Ockham, is that faith in the evaluative framework should not be blind, the scientist must be open to persuasion that they may be wrong. The view taken by Cox and Ince is that this can only occur after new data has appeared, but the vacuousness of this position is given by the example of the Black Swan and the way that science can avoid these lapses in prudence is by being open.

It might seem to be splitting hairs to argue in favour of science originating out of finance rather than magic, but I am of the opinion that creation myths have a critical role in how cultures view themselves. If science believes it emerges out of magic it will be forever associated with secret knowledge that enables the magician to control nature and convert base metal into gold. If we regard science as originating out of markets constructed social instruments, then it is natural that we think of science as being "just another" social construction and it is made more human, and possibly more relevant. Simultaneously, and this is my ultimate objective, we shall start observing markets from a scientific perspective, rather than having them hidden from public oversight by a veil of mystery and obscure incantations.

Now if I have been a bit obtuse, let me be more explicit. I worry about the views around the belief that "science" has a special status that makes it difficult to challenge was a contributory factor in the Credit Crisis. I will give an anecdote that a physical scientist would dismiss but social scientists would consider as evidence. After the collapse of British and American banks there was anger in the British and American banking community that the French had been engaged in exactly the same practices as the failed Anglo-Saxon banks, but had weathered the storm. I was party to a private conversation where a prominent mathematician who works for a prestigious French bank was asked about this complaint. The continentally trained mathematician pointed out that "we thought the models were probably wrong before the Crisis, we knew they were wrong during the Crisis, the view was the solvency of the bank should not be compromised on the basis of bad models". The UK/US banks can be seen as believing in the models before and during the Crisis, as a result, Lehmans, Bear Sterns BoS, Northern Rock and RBS no longer exist as independent entities where as Paribas and SocGen do. In a way, this aspect of the Credit Crisis can be seen as empirical evidence for a failure of Analytic Philosophy and a triumph for Continental Philosophy.

Despite the fact that I have spent an hour or so writing this piece, and you have spent time reading it, I do not think we should worry too much about the article by Cox&Ince. Neither of the authors is qualified to inform policy makers, and I suspect the likes of Andy Kerr advising on climate change, the scientists advising DEFRA on GMOs or the badger cull, or even my modest contribution to DECC's modelling efforts, believe that policy makers should be so confident of accepting scientific advice without it being tempered by political considerations. Cox & Ince, perhaps, should focus more on the distinction between science and entertainment rather than the relationship between science and politics.

Friday, 4 January 2013

I went "offline" over the Christmas break and so missed the fallout of Brian Cox and Robin Ince's article in the New Statesman Politicians must not elevate mere opinion over science. The essence of C&I's piece is that science exists as an "adjudicator above opinion", but C&I's "science" is narrowly defined, "science is a process, a series of structures that allow us, in as unbiased a way as possible, to test our assertions against Nature", essentially science is the set of phenomena that does not involve human interaction. They go on to say that

Science is the framework within which we reach conclusions about the natural world. These conclusions are always preliminary, always open to revision, but they are the best we can do. It is not logical to challenge the findings of science unless there are specific, evidence-based reasons for doing so. Elected politicians are free to disregard its findings and recommendations. Indeed, there may be good reasons for doing so. But they must understand in detail what they are disregarding, and be prepared to explain with precision why they chose to do so. It is not acceptable to see science as one among many acceptable “views”. Science is the only way we have of exploring nature, and nature exists outside of human structures.

This is a well structured argument. The five sentences are hard to disagree with, the paragraph gets our agreement, then it hits us with more controversial comments that to disregard scientific advice requires precise explanation. The final clause is telling "nature exists outside of human structures".

The problem I have with this assessment is that C&I claim science is essential in areas such as climate change, vaccines, GMOs and evolution. Now, apart from evolution, all these topics involve human interaction with nature: the issue about climate change is whether it is human induced; vaccines and GMOs are essentially compounds synthesised by humans and placed into "nature". While nature exists outside human structures, vaccines, GMOs and human-induced climate change cannot.

Brian Cox has Tweeted "Reviewed criticism of @robinince and my Christmas New Statesman, and concluded none is scientifically valid - so still time to get it :)". This gets to the heart of my problem with the C&I argument - they define what is "scientific" in such a way that it becomes impossible to argue a case against them "scientifically".

As a mathematician working in relation to finance, why should I care? Between 2006-2011 I was the "RCUK's Academic Fellow in Financial Mathematics", the Academic Fellowship scheme was initiated by the UK government in the aftermath of the MMR vaccine fiasco and the problems with the introduction of GMOs to establish 800 scientists who would act as advocates for their discipline in the event that discipline became a subject of the news.

In September 2008 I contacted the RCUK, my funders, about "what should I do", given that the world was in the grip of the Credit Crisis, the RCUK advised I contact the Science Media Centre, a publicly funded “venture working to promote the voices, stories and views of the scientific community to the national news media when science is in the headlines” to facilitate communication between mathematicians and science journalists on the credit crisis. I sent an e-mail to the SMC but did not here anything back. After I had been asked to appear on the BBC's Newsnight programme to discuss the science behind the Crisis, I phoned the SMC about my earlier contact. The SMC had decided that the Credit Crisis was not a science story, and therefore beyond their remit. This was really annoying because a highly political Press Officer, Fiona Fox, with no scientific credentials was telling me, someone who had a Physics BSc, had worked in a technology based industry, had a PhD in applied probability and was an established academic, what science was. The following February, I was contacted by the office of the Science Minister who asked me to collate the views of mathematicians n the Credit Crisis, they asked if I had any support from the SMC and I recounted the tale. Later that day I was contacted by the SMC to sort things out - some four months after the damage had been done. (see also one of my first posts Was the Credit Crisis a Science Story)

This experience with Ms Fox revealed to me some serious issues with British science. There are a hard-core of scientists, a vocal minority, who are convinced science is undervalued, because it is under-funded and the country is not run as a technocracy. They often complain that they are not taken seriously by society, while simultaneously withdrawing from addressing issues that are of concern to society: true science is about cosmology or particle physics, not about obesity or poverty.

What is perplexing is that, as many sociologists have pointed out, is that, for the likes of C&I, the one area that should not be open to rational examination (i.e. science) is their beloved "science".

Now lets get to the maths and finance. C&I are famous for being the presenters of the award winning BBC Radio Show The Infinite Monkey Cage. The title is an oblique reference to the idea that if you collect an infinite number of monkeys and typewriters they will eventually come up with the works of Shakespeare.To be precise, they will almost surely (a.s.) come up with the works of Shakespeare. The term a.s. is mathematical, a physicist like Cox would not worry about it, and it means that it is not certain that the monkeys would come up with the works, just that it is highly probable - the two are not the same as any financial practitioner is regularly reminded.

This distinction between a.s and certainty plays an important role in the philosophy of mathematics, in particular intuitionism or constructivism. L.E.J. Bouwer argued that statements like "there are a sequence of 100 9's in the decimal expansion of pi" (which as an irrational number has an infinite number of digits) can neither be proved to be true or false, in mathematics we cannot rely on the Law of Excluded Middle that assumes propositions are either true of false. If we cannot rely on truth/falsity of mathematical statements that apply to continuous phenomena (which involve infinite sets), then how can we rely on scientific statements to be true or false? The issue is that many of the scientists who worry about public disregard for science often do not undertake a rational assessment of the "scientific method" that they are so reliant on, and dismiss any questioning of it as "un-scientific" because it implicitly considers science as a human construct (which is the essence of my paper on Ethics and Finance). The story of Brouwer and the LEM is particularly pertinent here because Brouwer took a constructivist approach because he was a Marxist - it was a political act.

Brouwer was not isolated in his distrust of non-constructivist theories, Poincare, Borel and Lebesgue were equally circumspect of the approach. Again, what is pertinent here is that both Poincare and Borel were mathematicians who took an active role in political life and contemporary culture, they did not withdraw into academic cloisters and complain about society's disinterest in science.

Now the finance. C&I base their science on observation, data, and the predictive models constructed on the basis of the data. However there appears to be an assumption that "science" will come up with the right models, modulo the approximation problem, given the data. However this approach makes some omissions: what data is collected and why (science does not work by collecting reams of data in the hope something will drop out), data analysis is subjective (is climate data a hockey stick or a bath - see McIntyre&McKitrick, what does the data say?), models are human constructions.

Making these observations does not seem relevant to C&I, but they are crucial in modern finance, an arena of people competing to select and interpret data and develop the best models. It is a microcosm of good science, and for this reason it should be taken more seriously by the scientific establishment. Not least because modern finance is more relevant, and therefore more interesting, to the public than cosmology or theoretical physics.

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About Me

I am a Lecturer in Financial Mathematics at Heriot-Watt University in Edinburgh. Heriot-Watt was the first UK university to offer degrees in Actuarial Science and Financial Mathematics and is a leading UK research centre in the fields.

Between 2006-2011 I was the UK Research Council's Academic Fellow in Financial Mathematics and was involved in informing policy makers of mathematical aspects of the Credit Crisis.